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A fundamental problem associated with the task of network reconstruction from dynamical or behavioral data consists in determining the most appropriate model complexity in a manner that prevents overfitting, and produces an inferred network with a statistically justifiable number of edges. The status quo in this context is based on $L_{1}$ regularization combined with cross-validation. However, besides its high computational cost, this commonplace approach unnecessarily ties the promotion of sparsity with weight "shrinkage". This combination forces a trade-off between the bias introduced by shrinkage and the network sparsity, which often results in substantial overfitting even after cross-validation. In this work, we propose an alternative nonparametric regularization scheme based on hierarchical Bayesian inference and weight quantization, which does not rely on weight shrinkage to promote sparsity. Our approach follows the minimum description length (MDL) principle, and uncovers the weight distribution that allows for the most compression of the data, thus avoiding overfitting without requiring cross-validation. The latter property renders our approach substantially faster to employ, as it requires a single fit to the complete data. As a result, we have a principled and efficient inference scheme that can be used with a large variety of generative models, without requiring the number of edges to be known in advance. We also demonstrate that our scheme yields systematically increased accuracy in the reconstruction of both artificial and empirical networks. We highlight the use of our method with the reconstruction of interaction networks between microbial communities from large-scale abundance samples involving in the order of $10^{4}$ to $10^{5}$ species, and demonstrate how the inferred model can be used to predict the outcome of interventions in the system.

We consider the solution of systems of linear algebraic equations (SLAEs) with an ill- conditioned or degenerate exact matrix and an approximate right-hand side. An approach to solving such a problem is proposed and justified, which makes it possible to improve the conditionality of the SLAE matrix and, as a result, obtain an approximate solution that is stable to perturbations of the right hand side with higher accuracy than using other methods. The approach is implemented by an algorithm that uses so-called minimal pseudoinverse matrices. The results of numerical experiments are presented that confirm the theoretical provisions of the article.

This manuscript derives locally weighted ensemble Kalman methods from the point of view of ensemble-based function approximation. This is done by using pointwise evaluations to build up a local linear or quadratic approximation of a function, tapering off the effect of distant particles via local weighting. This introduces a candidate method (the locally weighted Ensemble Kalman method for inversion) with the motivation of combining some of the strengths of the particle filter (ability to cope with nonlinear maps and non-Gaussian distributions) and the Ensemble Kalman filter (no filter degeneracy).

This paper presents a regularized recursive identification algorithm with simultaneous on-line estimation of both the model parameters and the algorithms hyperparameters. A new kernel is proposed to facilitate the algorithm development. The performance of this novel scheme is compared with that of the recursive least squares algorithm in simulation.

We consider the problem of learning and using predictions for warm start algorithms with predictions. In this setting, an algorithm is given an instance of a problem, and a prediction of the solution. The runtime of the algorithm is bounded by the distance from the predicted solution to the true solution of the instance. Previous work has shown that when instances are drawn iid from some distribution, it is possible to learn an approximately optimal fixed prediction (Dinitz et al, NeurIPS 2021), and in the adversarial online case, it is possible to compete with the best fixed prediction in hindsight (Khodak et al, NeurIPS 2022). In this work we give competitive guarantees against stronger benchmarks that consider a set of $k$ predictions $\mathbf{P}$. That is, the "optimal offline cost" to solve an instance with respect to $\mathbf{P}$ is the distance from the true solution to the closest member of $\mathbf{P}$. This is analogous to the $k$-medians objective function. In the distributional setting, we show a simple strategy that incurs cost that is at most an $O(k)$ factor worse than the optimal offline cost. We then show a way to leverage learnable coarse information, in the form of partitions of the instance space into groups of "similar" instances, that allows us to potentially avoid this $O(k)$ factor. Finally, we consider an online version of the problem, where we compete against offline strategies that are allowed to maintain a moving set of $k$ predictions or "trajectories," and are charged for how much the predictions move. We give an algorithm that does at most $O(k^4 \ln^2 k)$ times as much work as any offline strategy of $k$ trajectories. This algorithm is deterministic (robust to an adaptive adversary), and oblivious to the setting of $k$. Thus the guarantee holds for all $k$ simultaneously.

This paper contributes to the study of optimal experimental design for Bayesian inverse problems governed by partial differential equations (PDEs). We derive estimates for the parametric regularity of multivariate double integration problems over high-dimensional parameter and data domains arising in Bayesian optimal design problems. We provide a detailed analysis for these double integration problems using two approaches: a full tensor product and a sparse tensor product combination of quasi-Monte Carlo (QMC) cubature rules over the parameter and data domains. Specifically, we show that the latter approach significantly improves the convergence rate, exhibiting performance comparable to that of QMC integration of a single high-dimensional integral. Furthermore, we numerically verify the predicted convergence rates for an elliptic PDE problem with an unknown diffusion coefficient in two spatial dimensions, offering empirical evidence supporting the theoretical results and highlighting practical applicability.

We propose a continuous approach for computing the pseudospectra of linear operators following a 'solve-then-discretize' strategy. Instead of taking a finite section approach or using a finite-dimensional matrix to approximate the operator of interest, the new method employs an operator analogue of the Lanczos process to work directly with operators and functions. The method is shown to be free of spectral pollution and spectral invisibility, fully adaptive, nearly optimal in accuracy, and well-conditioned. The advantages of the method are demonstrated by extensive numerical examples and comparison with the traditional method.

We propose an abstract conceptual framework for analysing complex security systems using a new notion of modes and mode transitions. A mode is an independent component of a system with its own objectives, monitoring data, algorithms, and scope and limits. The behaviour of a mode, including its transitions to other modes, is determined by interpretations of the mode's monitoring data in the light of its objectives and capabilities -- these interpretations we call beliefs. We formalise the conceptual framework mathematically and, by quantifying and visualising beliefs in higher-dimensional geometric spaces, we argue our models may help both design, analyse and explain systems. The mathematical models are based on simplicial complexes.

Matching on a low dimensional vector of scalar covariates consists of constructing groups of individuals in which each individual in a group is within a pre-specified distance from an individual in another group. However, matching in high dimensional spaces is more challenging because the distance can be sensitive to implementation details, caliper width, and measurement error of observations. To partially address these problems, we propose to use extensive sensitivity analyses and identify the main sources of variation and bias. We illustrate these concepts by examining the racial disparity in all-cause mortality in the US using the National Health and Nutrition Examination Survey (NHANES 2003-2006). In particular, we match African Americans to Caucasian Americans on age, gender, BMI and objectively measured physical activity (PA). PA is measured every minute using accelerometers for up to seven days and then transformed into an empirical distribution of all of the minute-level observations. The Wasserstein metric is used as the measure of distance between these participant-specific distributions.

We generalize McDiarmid's inequality for functions with bounded differences on a high probability set, using an extension argument. Those functions concentrate around their conditional expectations. We further extend the results to concentration in general metric spaces.